Methods of characterizing ion-exchanged chemically strengthened glasses containing lithium

ABSTRACT

Methods of characterizing ion-exchanged chemically strengthened glass containing lithium are disclosed. The methods allow for performing quality control of the stress profile in chemically strengthened Li-containing glasses having a surface stress spike produced in a potassium-containing salt, especially in a salt having both potassium and sodium. The method allows the measurement of the surface compression and the depth of the spike, and its contribution to the center tension, as well as the compression at the bottom of the spike, and the total center tension and calculation of the stress at the knee where the spike and the deep region of the stress profile intersect. The measurements are for a commercially important profile that is near-parabolic in shape in most of the interior of the substrate apart from the spike.

CROSS-REFERENCE TO RELATED APPLICATIONS

This Application is a continuation of U.S. patent application Ser. No.15/267,392, filed on Sep. 16, 2016, which claims the benefit of priorityto U.S. Provisional Patent Application Ser. No. 62/219,949, filed onSep. 17, 2015, and which is incorporated by reference herein.

FIELD

The present disclosure relates to chemically strengthened glass, and inparticular relates to methods of characterizing ion-exchanged chemicallyglasses containing lithium.

BACKGROUND

Chemically strengthened glasses are glasses that have undergone achemical modification to improve at least one strength-relatedcharacteristic, such as hardness, resistance to fracture, etc.Chemically strengthened glasses have found particular use as coverglasses for display-based electronic devices, especially hand-helddevices such as smart phones and tablets.

In one method, the chemical strengthening is achieved by an ion-exchangeprocess whereby ions in the glass matrix are replaced by externallyintroduced ions, e.g., from a molten bath. The strengthening generallyoccurs when the replacement ions are larger than the native ions (e.g.,Na+ ions replaced by K+ ions). The ion-exchange process gives rise to arefractive index profile that extends from the glass surface into theglass matrix. The refractive index profile has a depth-of-layer or DOLthat defines a size, thickness or “deepness” of the ion-diffusion layeras measured relative to the glass surface. The refractive index profilealso defines a number of stress-related characteristics, including astress profile, a surface stress, center tension, birefringence, etc.The refractive index profile defines an optical waveguide when theprofile meets certain criteria.

Recently, chemically strengthened glasses with a very large DOL (andmore particularly, a large depth of compression) have been shown to havesuperior resistance to fracture upon face drop on a hard rough surface.Glasses that contain lithium (“Li-containing glasses”) can allow forfast ion exchange (e.g., Li+ exchange with Na+ or K+) to obtain a largeDOL. Substantially parabolic stress profiles are easily obtained inLi-containing glasses, where the ion-exchange concentration profile ofNa+ connects in the central plane of the substrate, shrinking thetraditional central zone of the depth-invariant center tension to zeroor negligible thickness. The associated stress profiles have apredictable and large depth of compression, e.g., on the order of 20% ofthe sample thickness, and this depth of compression is quite robust withrespect to variations in the fabrication conditions.

A stress profile of particular commercial importance is a near-parabolic(substantially parabolic) profile that has a “spike” near the surface.The transition between the parabolic portion of the profile and thespike has a knee shape. The spike is particularly helpful in preventingfracture when the glass is subjected to force on its edge (e.g., adropped smart phone) or when the glass experiences significant bending.The spike can be achieved in Li-containing glasses by ion exchange in abath containing KNO₃. It is often preferred that the spike be obtainedin a bath having a mixture of KNO₃ and NaNO₃ so that Na+ ions are alsoexchanged. The Na+ ions diffuse faster than K+ ions and thus diffuse atleast an order of magnitude deeper than the K+ ions. Consequently, thedeeper portion of the profile is formed mainly by Na+ ions and theshallow portion of the profile is formed mainly by K+ ions.

In order for chemically strengthened Li-containing glasses to becommercially viable as cover glasses and for other applications, theirquality during manufacturing must be controlled to certainspecifications. This quality control depends in large part on theability to control the ion-exchange process during manufacturing, whichrequires the ability to quickly and non-destructively measure therefractive index (or stress) profiles, and particular the stress at theknee portion, called the “knee stress.”

Unfortunately, the quality control for glasses with spike stressprofiles is wanting due to the inability to adequately characterize theprofiles in a non-destructive manner. This inability has mademanufacturing of chemically strengthened Li-containing glasses difficultand has slowed the adoption of chemically strengthened Li-containingglasses in the market.

SUMMARY

An aspect of the disclosure is directed to methods of characterizingchemically strengthened Li-containing glasses having a surface stressspike, such as produced by an ion-exchange process (i.e., anin-diffusion of alkali ions) whereby in an example Li+ is exchanged withK+ and Na+ ions (i.e., Li+⇔K+, Na+). The methods result in a measurementof the surface compression and the depth of the spike, and itscontribution to the center tension, as well as the compression at thebottom of the spike, and the total center tension.

The method is preferably carried out to obtain a commercially importantstress profile, e.g., one that is near-parabolic in shape in most of theinterior of the substrate other than the spike adjacent the substratesurface. The spike is generally formed by the slower diffusion (and thusshallower) K+ ions while the substantially parabolic portion is formedby the faster (and thus deeper) diffusing Na+ ions. The method allowsfor confirmation that the profile has reached the near-parabolic regime,e.g., has a self-consistency check. The method can also includeperforming quality control of the glass samples being process. Suchquality control is important for a commercially viable manufacturingprocess.

The present disclosure provides a method for quality control of thestress profile in chemically strengthened Li-containing glasses having asurface stress spike produced in a potassium-containing salt, especiallyin a salt having both potassium and sodium. The method allows themeasurement of the surface compression and the depth of the spike, andits contribution to the center tension, as well as the compression atthe bottom of the spike, and the total center tension, for acommercially important profile that is near-parabolic in shape in mostof the interior of the substrate (apart from the spike). The methodallows to check that the profile has reached the near-parabolic regime,e.g., has a self-consistency check. The method provides a criticallyimportant tool for the quality control that is necessary for theadoption of lithium-containing glasses that allow the fabrication ofthese important profiles.

Prior art methods of measuring the stress level at the bottom of thespike (i.e., the knee stress) are limited by the relatively poorprecision of measuring the position of the critical-angle transition ofthe transverse electric (TE) angular coupling spectrum. This poorprecision is an inherent aspect of the TE transition, which is broad andhence appears blurred in the prism-coupling spectra. This lack ofsharpness causes the measured position of the mode lines to besusceptible to interference from nun-uniformity in the angulardistribution of the illumination (e.g., background non-uniformity), aswell as simply image noise.

Several of the methods disclosed herein avoid the need to measure theposition of the critical-angle of the TE transition precisely. In oneaspect of the method, the surface stress and the slope of the stress inthe spike are measured, as well as the depth (depth-of-layer, or DOL) ofthe spike, where the DOL is measured very precisely by using only thecritical-angle transition of the TM wave. This TM transition is sharperthan the TE transition and thus allows for a much more precisemeasurement. Thus, in an example of the method, the TE mode spectrum(and in particular the TE transition of the TE spectrum) is not used todetermine the DOL of the spike.

Knowing the surface stress and slope of the spike, and the depth of thespike (the aforementioned DOL), the stress at the bottom of the spike isdetermined, where the bottom of the spike occurs at the depth=DOL. Thisis the “knee stress” and is denoted herein as either CS_(knee) or CS_(k)or in the more general form σ_(knee). The rest of the calculation of thestress profile attributes then proceeds according to the prior artmethod.

A second method disclosed herein avoids a direct measurement of the kneestress and calculates the knee stress by using the birefringence of thelast guided mode common to both the TM and the TE polarization, and apreviously determined relationship between the birefringence of saidlast common guided mode and the stress at the knee. Advantage is takenof the generally better precision of measurement of the mode positionsin comparison to the precision of measurement of critical angle, and inparticular of the critical angle of the TE wave in the case of spikeddeep profiles in a Li-containing glass.

Advantages of the methods disclosed herein is that they arenon-destructive and can carried out with high-throughput and with highprecision to determine the critical parameters associated with thediffusion process in making chemically strengthened glasses. Thesecritical parameters include CS, depth of spike, estimate of thecompression depth, and frangibility status (based on an estimate of CTthat is provided by the method). Another advantage is that the methodscan be implemented with relatively modest software enhancements onexisting hardware used for quality control of the currently producedchemically strengthened glasses.

One major specific advantage of the new methods disclosed herein is asignificant improvement in the precision of the knee-stress estimate byavoiding the effects of large errors in the direct measurement of the TEcritical angle. This precision improvement is important because itallows for improved quality control of the chemically strengthened glassproduct.

The other advantage of the methods disclosed herein is an increase indomain of applicability of the methods, i.e., an increase in the size ofthe measurement process window. The prior art methods have processwindows or “sweet spots” for making measurements, where there was noleaky mode occurring in the vicinity of the critical-angle transitionfor the TM and TE spectra. Such a leaky mode causes significantdeformation of the angular distribution of intensity in the vicinity ofthe transition, and is a source of very significant and unacceptableerrors that are difficult to eliminate or effectively compensate for inrealistic situations.

In the first of the new methods, only the TM spectrum is required to befree of leaky-mode interference, which on average doubles the range ofthe sweet spot.

In both of the new methods, the effect of errors in the critical-anglemeasurement is significantly reduced because the critical angle is notused for a direct measurement of the knee stress. This leads to aneffective increase in the range of the sweet spot.

Additional features and advantages are set forth in the DetailedDescription that follows, and in part will be readily apparent to thoseskilled in the art from the description or recognized by practicing theembodiments as described in the written description and claims hereof,as well as the appended drawings. It is to be understood that both theforegoing general description and the following Detailed Description aremerely exemplary, and are intended to provide an overview or frameworkto understand the nature and character of the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings are included to provide a furtherunderstanding, and are incorporated in and constitute a part of thisspecification. The drawings illustrate one or more embodiment(s), andtogether with the Detailed Description serve to explain principles andoperation of the various embodiments. As such, the disclosure willbecome more fully understood from the following Detailed Description,taken in conjunction with the accompanying Figures, in which:

FIG. 1A is an elevated view of an example DIOX glass substrate in theform of a planar substrate;

FIG. 1B is a close-up cross-sectional view of the DIOX substrate of FIG.1A as taken in the x-z plane and that illustrates the doubleion-exchange process that takes place across the substrate surface andinto the body of the substrate;

FIG. 1C schematically illustrates the result of the DIOX process thatforms the DIOX substrate;

FIG. 2 is a representation of an example refractive index profile n(z)for the DIOX substrate illustrated in FIG. 1C;

FIGS. 3A and 3B are a photograph and a schematic diagram, respectively,of a mode spectrum based on a measured mode spectrum of a Li-containingglass formed by an ion-exchange process using a mixture of NaNO₃ andKNO₃, with the mode spectrum including a TM spectrum (top) and a TEspectrum (bottom), and also showing select profile measurementparameters pertinent to carrying out the methods disclosed herein;

FIG. 4 is a plot of the compressive stress CS (MPa) versus a normalizedposition coordinate z/L, showing the model stress profile (solid line)for a sample chemically strengthened Li-containing glass that hasundergone a K+ and Na+ ion exchange, wherein the dashed line curverepresents the model profile for Na+ diffusion only, noting that themodel profile has ion exchange taking place at two surfaces thatrespectively reside at z/T=−0.5 and +0.5;

FIG. 5 is a plot of the stress (MPa) versus a normalized positioncoordinate z/L showing separate plots for the spike portion, the longdiffused parabolic portion and the spike plus parabolic portion of thestress profile;

FIGS. 6A and 6B are a photograph and a schematic diagram, respectively,of a mode spectrum based on a measured mode spectrum and showing the TEand TM mode spectra for an example chemically strengthened Li-containingglass sample;

FIGS. 6C and 6D are plots of the intensity I versus distance x′ alongthe mode spectrum for the mode lines or fringes of TE and TM modespectra, respectively, of FIG. 6A;

FIGS. 7A through 7C show a series of mode spectra as measured on anexample chemically strengthened glass substrate, wherein themeasurements were made at different diffusion times;

FIGS. 8A and 8B are plots of measured DOL(μm) vs. time, where region R1shows the measurement based on two fringes (TM mode) at 589 nm, regionR2 shows the transition between using three fringes (TM mode) at 589 nm,and region R3 shows the measurement using three fringes (TM mode) at 589nm;

FIGS. 9A, 9B and FIGS. 10A, 10B are plots of the measured compressivestress CS (MPa) versus time in hours (hrs) and showing the same regionsR1, R2 and R3 as FIGS. 8A and 8B;

FIG. 11 sets forth a Table 1 that shows the computation of profileparameters for a number of different glass samples;

FIG. 12 shows a reduced range and reduced dependence of CS_(k) on CS formeasured samples covering a range of DOL after applying an improvedalgorithm for slope extraction (fitted curve B) based on using three ormore optical modes per polarization as opposed to two optical modes perpolarization (fitted curve A);

FIG. 13 shows a Table 3 that includes the calculated effective indicesof the 3 guided modes for both the TM and TE polarizations for severaldifferent assumed values of CS_(k), CS, and DOL for an example waveguideformed in a glass substrate;

FIG. 14A shows a dependence of a knee-stress scaling factor K₃ on thecompressive stress CS for an example simulated ion exchangedLi-containing glass, for the purpose of applying a dynamically adjustedCS-dependent factor for correcting systematic error of CS_(k)measurement in one embodiment, the systematic error resulting from K₃being assumed constant;

FIG. 14B shows the derivative of scaling factor K₃ with respect to CS(dK₃/dCS) and the minor dependence of that derivative on the CS value,for the purposes of correcting systematic error in CS_(k) in oneembodiment where the systematic error results from K₃ being assumedconstant;

FIG. 15 shows dependence of the scaling factor K₃ on DOL, for simulatedprofiles covering a range of DOL, for the purpose of correcting asystematic error in CS_(k) resulting from assuming that K₃ is constantand independent of DOL; and

FIG. 16 shows a derivative of the scaling factor K₃ with respect to DOL(dK₃/dDOL) having a region of relatively small and little changingderivative, and a region of fast-changing derivative, growingsubstantially in absolute value.

DETAILED DESCRIPTION

Reference is now made in detail to various embodiments of thedisclosure, examples of which are illustrated in the accompanyingdrawings. Whenever possible, the same or like reference numbers andsymbols are used throughout the drawings to refer to the same or likeparts. The drawings are not necessarily to scale, and one skilled in theart will recognize where the drawings have been simplified to illustratethe key aspects of the disclosure.

The claims as set forth below are incorporated into and constitute partof this Detailed Description.

FIG. 1A is an elevated view an example glass substrate in the form of aplanar ion-exchanged substrate 20 that has a body 21 and a (top) surface22, wherein the body has a base (bulk) refractive index n_(s), a surfacerefractive index n₀ and a thickness T in the z-direction. FIG. 1B is aclose-up cross-sectional view of ion-exchanged substrate 20 as taken inthe y-z plane and illustrates an example double ion-exchange (DIOX)process that takes place across surface 22 and into body 21 in thez-direction.

In the DIOX process discussed in connection the method disclosed herein,two different types of ions Na+ and K+ replace another different ion Li+that is part of the glass body 21. The Na+ and K+ ions can be introducedinto the glass body 21 either sequentially or concurrently using knownion-exchange techniques. As noted above, the Na+ ions diffuse fasterthan the K+ ions and thus go deeper into the glass body 21. This has aneffect on the resulting refractive index profile and stress profile, asdiscussed below.

FIG. 1C is a schematic diagram of the resulting DIOX process, and FIG. 2is a representation of an example refractive index profile n(z) forsubstrate 20 having undergone the DIOX process and having a refractiveindex profile such as illustrated in FIG. 1C. The corresponding stressprofile can be represented by σ(z). The refractive index profile n(z)includes a first “spike” SP associated with a region R1 associated withthe shallower ion-exchange (K+ ions) and that has a depth D1 into body21 that defines a “depth-of-layer for the spike” also denotedhereinafter as DOL_(sp). The refractive index profile n(z) also includesa second region R2 associated with the deeper ion-exchange (Na+ ions)and that has a depth D2 that defines the depth-of-layer (DOL) and alsodenoted DOL_(p). In an example, the portion of the refractive indexprofile n(z) in second region R2 is denoted PP because it has aparabolic shape or generally a power-law shape. The spike SP andpower-law profile PP intersect at a location KN that has the shape of aknee.

The deeper second region R2 may be produced in practice prior to theshallower region. The region R1 is adjacent substrate surface 22 and isrelatively steep and shallow, whereas region R2 is less steep andextends relatively deep into the substrate to the aforementioned depthD2. In an example, region R1 has a maximum refractive index n₀ atsubstrate surface 22 and steeply tapers off to an intermediate indexn_(i), while region R2 tapers more gradually from the intermediate indexdown to the substrate (bulk) refractive index n_(s). The portion of therefractive index profile n(z) for region R1 represents spike SP in therefractive index having a depth DOS.

FIG. 3A is photograph of example measured mode spectrum 50 and FIG. 3Bis a schematic diagram of the measured mode spectrum of FIG. 3A, whichis for Li-containing glass formed by an ion-exchange process using amixture of NaNO₃ and KNO₃. The mode spectrum 50 includes TM and TEspectra 50TM and 50TE (upper and lower portions, respectively) withrespective mode lines 52TM and 52TE that represent higher-order modes.The lower-order modes lines in the TM and TE mode spectra 50TM and 50TEare tightly bunched together and are shown as respective solid blackregions denoted as 54TM and 54TE. The glass type for the measured samplewas 196HLS with a fictive temperature of 638° C. The glass sample wassubjected to a Li+⇔K+, Na+ ion-exchange process by placing the glasssample in a bath having 60 wt % KNO₃ and 40 wt % NaNO₃ at 390° C. for 3hours.

As is known in the art, the fringes or mode lines 52TM and 52TE in themode spectrum can be used to calculate surface compression or“compressive stress” CS and depth of layer DOL associated with anion-exchange layer that forms an optical waveguide. In the presentexample, the mode spectrum 50 on which FIGS. 3A and 3B are based wasobtained using a commercially available prism-coupling system, namelythe FSM6000L surface stress meter (“FSM system”), available from LuceoCo., Ltd. of Tokyo, Japan. Example prism-coupling systems suitable foruse for carrying out the methods disclosed herein are also described inU.S. Patent Applications No. 2014/0368808 and 2015/0066393, which areincorporated by reference herein.

The measured values of CS and DOL were 575 MPa and 4.5 microns,respectively. These are the parameters of the K+ enriched layer or spikeregion R1 adjacent sample surface 22 (see FIG. 2). The vertical dashedlines on the left hand side of the spectrum of FIG. 3A show positions inthe spectrum that correspond to the surface index, one for TM, and onefor TE. The difference in these positions, as indicated by the blackarrows, is proportional to the surface stress or compressive stress CS.One of the black arrows in FIG. 3A is denoted CS_(tot), while the otheris denoted CS_(knee) or CS_(k) or and are discussed below. These valuesare used in the calculation of DOL.

In the mode spectrum 50 for a chemically strengthened Li-containingglass having undergone a (Li+⇔K+, Na+) ion exchange, the relativepositions of the TM and TE mode spectra 50TM and 50TE are shifted. Thisshift can be measured by the relative positions of the last (i.e.,left-most) fringes 52TM and 52TE, which correspond to the highest-orderguided modes. As noted above, this shift is denoted CS_(tot) in FIGS. 3Aand 3B and is proportional to the compressive stress CS at the depth atwhich the K+ concentration in spike region R1 decreases approximately tothe constant-level concentration originally in the substrate (e.g., thespatially constant concentration in the glass matrix that makes upsubstrate body 21).

The effective index of the transition corresponds to the effective indexthat occurs at the depth of a characteristic “knee” or transition KN inthe stress profile, and is denoted in FIGS. 3A and 3B for the TM and TEmode spectra 50TM and 50TE, respectively. The shift of the transitionbetween the TE and TM spectra is proportional to the compressive stressat the depth of the knee KN and denoted CS_(knee) in FIGS. 3A and 3B.

The direct measurement of the knee stress CS_(knee) from thebirefringence of the critical-angle intensity transition of the TE andTM mode lines 52TE and 52TM presents some problems. One problem is dueto shifting of the apparent position of the transition when a leaky modeor a guided mode has effective index very close to the indexcorresponding to the critical angle. For example, the broader darkfringe can occur approximately at the same location as thecritical-angle transition in the upper half of the combined spectra ofFIGS. 3A and 3B.

FIGS. 6A and 6B are a photograph and a schematic diagram, respectively,of an example measured mode spectrum and showing the TE and TM modespectra for an example chemically strengthened Li-containing glasssample. FIGS. 6C and 6D are plots of the intensity I versus distance x′(I(x′)) along the mode spectrum for the mode lines or fringes 52TE and52TM of TE and TM mode spectra, respectively, of FIG. 6A. The intensityprofiles show the detection of the position x′ of the pixelscorresponding to the positions of the fringe and the transition betweenthe spike region and the continuum (last peak) for the TM and TE modes.The position in pixels is essentially a measurement of the index ofrefraction of the modes and the transition region.

Avoiding the aforementioned shift-induced error requires that both theupper and lower spectra (i.e., the TM and TE spectra 50TM and 50TE) thehave a fractional part of the mode number between about 0.2 and 0.6,which is quite restrictive. In addition, even when this condition issatisfied, the measurement of the critical angle for the TE spectrum isnot very precise due to a relatively blurry TE intensity transition.Note for example how the critical-angle transition in the bottom half ofFIGS. 3A and 3B is relatively broad, rather than sharp; in contrast, thecritical-angle TM transition shown in FIG. 6A is narrow (sharp) eventhough there is no dark fringe close to it.

The methods disclosed herein utilize measurements of the fringe spectrumprovided by the potassium penetration resulting from ion exchange, alongwith the position of the intensity transition in the TM spectrum (e.g.,transition from total internal reflection (TIR) to partial reflection)relative to the positions of the TM fringes. These measurements can becombined and used for effective quality control of a family of stressprofiles that help enable superior resistance to fracture during facedrops. The profiles of this family are similar in shape to a power-lawprofile with a spike.

The spike SP is a near-surface region that has a small thickness whencompared to the substrate thickness. For example, the spike may be 10 μmdeep, while the substrate may be 800 μm thick. The spike may have ashape similar to erfc-shape, but may also be similar to a linear depthdistribution, Gaussian depth distribution, or another distribution. Themain features of the spike are that it is a relatively shallowdistribution and provides substantial increase of surface compressionover the level of compression at the bottom (deepest end) of the spike,which ends at knee KN.

FIG. 4 shows a model stress profile having a power-law portion PP inregion R2 and surface spike SP in region R1, as shown in FIG. 2. For thepurposes of the present disclosure, the assumed convention is thatcompressive stress is positive and tensile stress is negative. The modelprofile of FIG. 4 has a linear spike SP added on top of a deep quadraticprofile for power-law portion PP.

Another feature of the spike SP in FIG. 4 is also recognized from FIG.2, namely that the typical slope of the stress distribution in the spikeSP is significantly higher than the typical slope in the power-lawportion PP of the profile. In an example, the power-law portion PP isassumed to be well-described by a power-function of the distance fromthe center-plane of the glass substrate, with a power exponent in therange between about 1.5 and about 4.5. In an example, this auxiliarypower-law portion PP can be assumed near-parabolic, and approximated asparabolic for the purposes of quality-control measurements.

In one embodiment of the method, the CS_(SP) and DOL_(SP) of the spikeSP are measured using a traditional FSM measurement. For increasedprecision of the DOL measurement, it may be preferred that the DOL_(SP)of the spike be measured using the TM spectrum only, as thecritical-angle transition in the example Li-containing glasses exchangedin mixtures of Na and K is substantially sharper and less prone tomeasurement errors. Note that in the present disclosure thedenominations DOL and DOL_(SP) are used interchangeably to refer to thesame quantity, namely, the depth of layer of the K-enriched near-surfacespike layer having high compressive stress CS_(SP).

A center tension CT contribution of the spike is calculated using theequation

${CT}_{sp} = \frac{{CS}_{sp} \times {DOL}_{sp}}{T - {DOL}_{sp}}$

where T is the sample thickness (see FIG. 1). The contribution to thecenter tension CT of the Na profile can be associated with the measuredknee stress σ_(knee). A crude estimate can be found by assuming that thesurface stress of the auxiliary profile used to describe the stressproduced by the Na distribution is approximately the same as the stressat the knee. Thus, we have:

${CT}_{p} \approx \frac{\sigma_{knee}}{p}$where σ_(knee) is the stress at the knee of the profile, e.g., at thebottom of the spike and is given by:

$\sigma_{knee} = \frac{\left( {n_{crit}^{TE} - n_{crit}^{TM}} \right)}{SOC}$where n_(crit) ^(TE) and n_(crit) ^(TM) are the effective indices of thecritical-angle intensity transitions as illustrated nor on FIGS. 3A and3B. The parameter SOC is the stress-optic coefficient. The numerator inthe above expression can be defines as the birefringence BR, in whichcase the equation reads:σ_(knee)=CS_(knee) =BR/SOC.This equation can also be written more generally asσ_(knee)=CS_(knee)=(CFD)(BR)/SOCwhere CFD is calibration factor between 0.5 and 1.5 that accounts forsystematic offsets between the recovered critical-angle values having todo with fundamentally different slopes of the TM and TE intensitytransitions, different shape of the TM and TE index profiles in thevicinity of the knee, and specifics of the method by which the locationof the intensity transition is identified. As noted above, theparameters σ_(knee), CS_(knee), CS_(k) and CS_(K) all refer to the samequantity, namely, the knee stress.

As illustrated by the dashed line curve in FIG. 4, the assumed power-lawor power-law profile for power-law portion PP may be taken as anauxiliary profile that does not include the spike, but extends thepower-law or parabolic shape all the way to the surfaces of the sample.This auxiliary profile is force-balanced, having its own compressivetension CT, and is hence shifted vertically from the power-law portionof the model spiked power-law profile.

Auxiliary Power-Law Profile Relationships

A detailed description of the relationships that hold for the auxiliarypower-law profile is now provided, as well as the associated method ofusing them to calculate the parameters of the model spiked profile forthe purposes of quality control.

The auxiliary power-law profile provides the stress as a function ofdistance z from the center.

${\sigma_{p}(z)} = {{CT}_{p} - {\sigma_{0}\left( \frac{z}{0.5T} \right)}^{p}}$${CT}_{p} = \frac{{CS}_{p}}{p}$${DOC}_{p} = {0.5{T\left( {1 - \frac{1}{\left( {1 + p} \right)^{\frac{1}{p}}}} \right)}}$

The spiked profile has a somewhat smaller depth of compression DOC givenby the expressions

${D\; O\; C} = {{0.5{T\left( {1 - \left( \frac{{CT}_{tot}}{\left( {1 + p} \right){CT}_{p}} \right)^{\frac{1}{p}}} \right)}} \equiv {0.5{T\left( {1 - \left( {\frac{1}{1 + p}\left( {1 + \frac{{CT}_{sp}}{{CT}_{p}}} \right)} \right)^{\frac{1}{p}}} \right)}}}$The depth of compression DOC of the spiked profile is smaller than thatof the auxiliary power profile by approximately:

${\Delta\;{DOC}} \approx {{- \frac{0.5t}{\left( {1 + p} \right)^{\frac{1}{p}}}} \times \frac{{CT}_{sp}}{{pCT}_{p}}}$The change in the depth of compression DOC caused by the spike in theprofile can be normalized to the compressive tension CT of the auxiliarypower profile as follows:

${\frac{\Delta\; D\; O\; C}{{DOC}^{aux}} \approx {- \frac{\frac{0.5T}{\left( {1 + p} \right)^{\frac{1}{p}}} \times \frac{{CT}_{sp}}{p\;{CT}_{p}}}{0.5{T\left( {1 - \frac{1}{\left( {1 + p} \right)^{\frac{1}{p}}}} \right)}}}} = {- \frac{{CT}_{sp}}{{p\left( {\left( {1 + p} \right)^{\frac{1}{p}} - 1} \right)}{CT}_{p}}}$

In the specific example of a parabolic auxiliary profile, the followingrelationships hold:

-   -   The auxiliary profile has a compression depth DOC_(par) given        by:

${DOC}_{par} = {{0.5{T\left( {1 - \frac{1}{\sqrt{3}}} \right)}} \approx {0.2113T}}$

-   -   The total center tension CT_(tot) of the profile equals the sum        of spike center tension CT_(sp) and the parabolic portion center        tension CT_(p):        CT_(tot)=CT_(p)+CT_(sp)    -   The depth of compression DOC of the spiked power-law profile can        be calculated by using the expression:

${DOC} = {{0.5{T\left( {1 - \sqrt{\frac{{CT}_{tot}}{3{CT}_{par}}}} \right)}} = {{0.5{T\left( {1 - \sqrt{\frac{1}{3}\left( {1 + \frac{{CT}_{sp}}{{CT}_{par}}} \right)}} \right)}} \approx {{DOC}_{par}\left( {1 - \frac{{CT}_{sp}}{2{{CT}_{par}\left( {\sqrt{3} - 1} \right)}}} \right)} \approx {{DOC}_{par} - {\frac{0.5T}{\sqrt{3}}\frac{{CT}_{sp}}{2{CT}_{par}}}}}}$

The approximate expressions at the end of the above equation are validwhen the CT contribution of the spike is significantly smaller than theCT contribution of the auxiliary profile (i.e., the parabolic portionPP).

Example Method Based on Approximation

An example method of quality control utilizes an approximation approachthat includes a measurement of the mode spectrum due to the spike. Themethod then includes estimating a contribution of the spike to thecenter tension CT by estimating a compression at the knee KN of theprofile and subtracting that knee compression from the surfacecompression in the calculation of the spike contribution to the centertension. The method then includes estimating a contribution to thecenter tension CT due to the deep power-law profile portion PP excludingthe spike, also taking advantage of the estimated knee stress. Themethod then includes finding the total center tension CT_(tot) as a sumof the contributions of the auxiliary deep power-law profile and of thespike, i.e., CT_(tot)=CT_(sp)+CT_(p). In general, the CT contribution ofthe deep portion may be denominated CT_(deep), which can beinterchangeably used with CT_(p) when the deep portion is represented ashaving a a power-law shape.

In addition, the method can include estimating the compression depth DOCof the profile by using an exact formula for the model profile, or anapproximate formula that gives the DOC as the DOC of an auxiliarypower-law profile less a small DOC reduction due to the spike, i.e.,DOC=DOC_(p)+ΔDOC_(SP) (in the mathematical formula a negative ΔDOC_(SP)is added to DOC_(p)). Note also that ΔDOC_(SP) is sometimes labeledsimply as A DOC in the present disclosure, as only the shift in DOC thatis due to the spike is considered in this disclosure.

In one example of the method, the DOL of the spike SP is used to verifythat the power-law portion PP of the profile (see FIG. 4) is in a regimethat is well represented by the power-law profile shape. In particular,as the DOL of the spike increases, the penetration of Na increasesapproximately in proportion to the DOL of the spike. Thus, for a glasssubstrate where simultaneous in-diffusion of K and Na is used, a minimumspike DOL_(SP) can be set for any particular glass thickness, abovewhich the deep portion of the profile can be considered parabolic. Inanother example, an upper limit of the DOL_(SP) may also be imposed, toexclude physical profiles that start to deviate substantially from theassumed power-law model.

More Precise Method

The above-described method is based on approximation and is thus asomewhat more simplified version of a more precise method. Thesimplification incurs only a minor error when the CT contribution of thespike is much smaller than the CT contribution of the auxiliarypower-law profile. The CT contribution of the spike shifts the deeppower-law portion PP vertically by the amount CT_(sp) relative to theauxiliary power-law profile. As a result, the compression at the knee ofthe model spiked profile is actually smaller than the compression of theauxiliary profile at the knee depth by the amount CT_(sp).

Furthermore, there is a minor change in compression of the auxiliarypower-law profile between the surface and the depth of the knee, and,for a force-balanced power-law profile the CT is actually equal to

$\frac{{CS}_{p}}{p}.$

The following represents an example of a more precise method fordetermining the parameters of the model spiked power-law profile fromthe mode spectrum as obtained from prism-coupling measurements of achemically strengthened glass sample:

-   -   a) Calculate preliminary

${CT}_{sp}^{(0)} = \frac{\left( {{CS}_{tot} - {CS}_{knee}} \right) \times {DOL}_{sp}}{T - {DOL}_{sp}}$

-   -   b) Calculate preliminary surface compression of the auxiliary        profile

${CS}_{p}^{(0)} = {p\frac{{CS}_{knee} + {CT}_{sp}^{(0)}}{{\left( {p + 1} \right)\left\lbrack {1 - \frac{2{DOL}_{sp}}{T}} \right\rbrack}^{p} - 1}}$

-   -   c) (Optional alternative to steps 4, 5, and 6) Calculate        preliminary

${CT}_{p}^{(0)} = {\frac{{CS}_{p}^{(0)}}{p} = \frac{{CS}_{knee} + {CT}_{sp}^{(0)}}{{\left( {p + 1} \right)\left\lbrack {1 - \frac{2{DOL}_{sp}}{T}} \right\rbrack}^{p} - 1}}$and CT_(tot) ⁽⁰⁾=CT_(p) ⁽⁰⁾+CT_(sp) ⁽⁰⁾

-   -   d) Calculate more precise

${CT}_{sp}^{(1)} = \frac{\left( {{CS}_{tot} - {CS}_{p}^{(0)}} \right) \times {DOL}_{sp}}{T - {DOL}_{sp}}$

-   -   e) Calculate more precise

${{CS}_{p}^{(1)} = {p\frac{{CS}_{knee} + {CT}_{sp}^{(1)}}{{\left( {p + 1} \right)\left\lbrack {1 - \frac{2{DOL}_{sp}}{T}} \right\rbrack}^{p} - 1}}},{and}$${CT}_{p}^{(1)} = {\frac{{CS}_{p}^{(1)}}{p} = \frac{{CS}_{knee} + {CT}_{sp}^{(1)}}{{\left( {p + 1} \right)\left\lbrack {1 - \frac{2{DOL}_{sp}}{T}} \right\rbrack}^{p} - 1}}$

-   -   f) Calculate more precise CT_(tot) ⁽⁰⁾=CT_(p) ⁽⁰⁾+CT_(sp) ⁽⁰⁾    -   g) (Optional; usually unnecessary)—can continue iteration,        finding more and more precise values for CT_(SP) and CS_(par)        until desired level of convergence or precision. More than one        iteration would rarely be needed. More than one iteration may be        useful in relatively thin substrates in which the depth of the        spike may represent more than about 3% of the substrate        thickness.    -   h) (Optional) Determine depth of compression of the profile, for        example using one of the forms of the equation:

${DOC} = {{0.5{T\left( {1 - \left( \frac{{CT}_{tot}}{\left( {1 + p} \right){CT}_{p}} \right)^{\frac{1}{p}\;}} \right)}} \equiv {0.5{T\left( {1 - \left( {\frac{1}{1 + p}\left( {1 + \frac{{CT}_{sp}}{{CT}_{p}}} \right)} \right)^{\frac{1}{p}\;}} \right)}}}$

The above-described method allows for the application of the genericauxiliary power-law profile for the QC of a spiked double-ion-exchangedprofile having a stress distribution reasonably well described by aspiked power-law profile model. The method avoids a direct measurementof the knee stress. Instead of directly measuring n_(crit) ^(TE) toevaluate the knee stress from the earlier described equation,

${\sigma_{knee} = \frac{\left( {n_{crit}^{TE} - n_{crit}^{TM}} \right)}{SOC}},$the knee stress is found by observing that it occurs at a depth equal tothe penetration of the spiking ion, e.g., at a depth of spike DOL_(sp).CS_(knee)≡σ_(knee)=σ(depth=DOL_(sp)).

The above strict definition of the knee stress is most easily understoodfor the case where the profile has an abrupt change in slope at thelocation of the knee. In practice, most profiles change slope gradually,although fast, in the vicinity of depth=DOL_(SP), and σ_(knee) occursapproximately at depth=DOL_(SP) as measured from the mode spectrum.Hence, in the calculation of σ_(knee) often a calibration factor ofmagnitude comparable to 1 is used, in part to account for differencesbetween the continuous distribution of stress and the abrupt change instress slope in a simple explicit description of a model having a steeplinear truncated stress spike connected to a deep region of slowlyvarying stress.

The surface stress and its slope are obtained from the prism-couplingmeasurements of the effective indices of the TM and TE modes confined inthe depth region of the spike by a measurement of the CS, the stressslope s_(σ) and DOL of the spike.

The surface stress and the slope of a linear spike can be found usingthe following analysis: Using the WKB approximation the turning pointsx₁ and x₂ of the two lowest-order modes in an optical waveguide can befound using the relations

$x_{1} = {\frac{9}{16}\frac{\lambda}{\sqrt{n_{0}^{2} - n_{1}^{2}}}}$$x_{2} = {\frac{21}{16}\frac{\lambda}{\sqrt{n_{0}^{2} - n_{2}^{2}}}}$where n₀ is the surface index of the profile having linearly decreasingwith depth dielectric susceptibility, n₁ is the index of thelowest-order mode, n₂ is the effective index of the second-lowest-ordermode, and λ is the optical wavelength. The surface index of the linearprofile is found from the same first two modes by the relation:n ₀ ² ≡n _(surf) ² ≈n ₁ ²+1.317(n ₁ ² −n ₂ ²)

For profiles having n₁−n₂«n₁, an even simpler relation can be used:n ₀ ≡n _(surf) ≈n ₁+1.3(n ₁ −n ₂)

The index slope of each of the TM and TE index profiles associated withthe stress profile of the spike is then given by:

$s_{n} = {\frac{n_{1} - n_{2}}{x_{1} - x_{2}}.}$The above relations for the surface index and the index slope of thelinear profile can be applied for both the TM and TE mode spectra, toobtain the TM and TE surface indices n_(surf) ^(TM) and n_(surf) ^(TE),and the TM and TE profile index slopes s_(n) ^(TM) and s_(n) ^(TE). Fromthese, the surface stress CS, and the stress slope s_(σ) can beobtained:

${CS} = \frac{n_{surf}^{TE} - n_{surf}^{TM}}{SOC}$$s_{\sigma} = \frac{s_{n}^{TE} - s_{n}^{TM}}{SOC}$where as noted above, SOC stands for stress-optic coefficient. Note thatwhen more than two guided modes are supported in either the TM or TEpolarization, or both, then the precision of the slope measurement canbe improved by taking advantage of the measured effective indices ofmore than two modes per polarization, by using a linear regression toassociate the measured effective indices of multiple modes with a singleindex slope for each polarization.

There is now one step left to obtain the knee stress, namely ameasurement of the spike depth OL_(sp), which is obtained by analysis ofthe TM spectrum. The index space between the highest-order guided modeand the index corresponding to the TM critical angle is assigned afraction of a mode based on what fraction it represents of the spacingof the previous two modes, and, if desired for higher precision, on howmany guided modes are guided. This type of DOL calculation is routinelydone by the FSM-6000 instrument.

Finally, the depth of the spike is given by the formula:

${DOL}_{sp} = {\frac{3}{4}\lambda\frac{N - \frac{1}{4}}{\sqrt{2{n_{av}\left( {n_{surf} - n_{crit}} \right)}}}}$where N is the number of guided TM modes, including the fraction of amode assigned to the space between the last guided mode and the criticalindex n_(crit) of the intensity transition, A is the measurementwavelength, and n_(crit) is the effective index corresponding to thecritical angle in the TM spectrum, indicated as n_(crit) ^(TM) in FIGS.3A and 3B.

With DOL_(sp) measured with good precision from the TM couplingspectrum, the knee stress CS_(knee) at the bottom of the spike is foundusing the relationship:CS_(knee)≡σ_(knee)≡σ_(sp)(x=DOL_(sp))=CS+s _(σ)×DOL_(sp)

Accounting for systematic differences between real profiles in thevicinity of the knee point, and the assumed model for the spike shape,the knee stress can be found by the following more general relationship:CS_(knee)≡σ_(knee)≡σ_(sp)(x=DOL_(sp))=CS+KCF×s _(σ)×DOL_(sp)where the knee calibration factor KCF is usually between 0.2 and 2, andserves to account for the difference in shape between a real spikedistribution and the assumed model of the spike shape, as well as theparticular way that the DOL_(sp) is calculated from the mode spectrum.For example, a commonly used equation for the surface index isn ₀ ≡n _(surf) ≈n ₁+0.9(n ₁ −n ₂).which uses a factor of 0.9 instead of the factor 1.317 which is accuratefor linear spikes. When the formula for surface index with a factor of0.9 is used, the resulting calculated DOL appears higher than the purelylinear-spike DOL.

This improved method of measurement of the knee stress by use of aprecise measurement of DOL_(sp), when used in the approximate algorithmor in the more precise in the iterative algorithm for extraction of theparameters of the spiked deep profile described above for the generalpower-law auxiliary profile (or in the previous disclosure for thequadratic auxiliary profile), provides a quality-control method withimproved precision of the estimate of CT for frangibility control. Theknee stress is by itself an important parameter of glass strength andthe precision improvement of that parameter is also of value. Theimproved method also increases the breadth of the sweet spot formeasurement typically by a factor of two or even more.

In another embodiment involving indirect measurement of the knee stress,the method makes use of a strong correlation between the knee stress andthe birefringence of the last guided mode of the spike. When the spikeCS and DOL are kept in very narrow respective ranges, then a strongcorrelation forms between the sought after knee stress and thedifference in the effective index between the last guided TM mode andthe last guided TE mode of the spike.

The method exploits the birefringence of the last guided mode of thespectrum acquired by the prism coupler for quality control (QC)measurements. Here we will use formulas for a generic power profile withexponent ‘n’. For a power-law profile n=2, for cubic n=3 but alsofractional profiles like n=2.37 is possible for making the equationsgeneric. In the present disclosure, when n refers to a power of theprofile, it has the same meaning as p which is also used to denominatethe power of the auxiliary deep profile.

Using the power (parabolic for n=2 in this case) auxiliary profile,illustrated with the help of FIG. 5, the following representation isused for the force-balanced profile:

$\begin{matrix}{{\sigma\left( \frac{z}{L} \right)} = {{CS}\left\lbrack {{\left( {1 + \frac{1}{n}} \right){\left( {{2\frac{z}{L}} - 1} \right)}^{n}} - \frac{1}{n}} \right\rbrack}} & (1)\end{matrix}$where L is the thickness. The depth of layer DOL_(deep) of the deep partof this power profile with exponent ‘n’ is given by

$\begin{matrix}{\frac{{DOC}_{deep}}{L} = {0.5\left( {1 - \frac{1}{\left( {1 + n} \right)^{\frac{1}{n}}}} \right)}} & (2)\end{matrix}$

The FSM measures FSM_DOL of the spike as approximately the diffusiondepth given by 2√{square root over (D·τ)} where D is the diffusioncoefficient and τ is the time of diffusion.

For a spike with the shape of erfc-function, it is empirically foundthat the knee stress can be assumed to occur at a depth of˜K₁×FSM_DOL=1.25×FSM_DOL, such that most of the stress-area of the spiketo be included in the CT calculation.

One can get an approximate equation for the ΔCT_(spike) due to the spikecontribution. Here, K₁ is an empirical factor set at 1.25 for thisparticular case. The factor K₁ serves to compensate for nonzero residualstress contributed by the tail of the spike at depth=FSM_DOL byadjusting the point at which the knee stress is estimated.

$\begin{matrix}{{\Delta\;{CT}_{spike}} = \frac{\left( {{\sigma_{2}^{\prime}(0)} - {\sigma_{2}^{\prime}\left( {K_{1} \times {FSM\_ DOL}} \right)}} \right) \times K_{1} \times {FSM\_ DOL}}{T - {K_{1} \times {FSM\_ DOL}}}} & (3)\end{matrix}$

The point σ₂′(x₁×FSM_DOL)=σ₂′(1.25×FSM_DOL) is very close to the CSbetween the transition between guided modes and continuum in the spikedlithium glass samples. This point is called the CS_(knee) as shown inFIGS. 3A and 3B as a reasonable approximation. It is also reasonable toapproximate the stress and the offset due to the contribution of theCT_(spike) deeper inside the glass.

Since the power-law profile will be slow varying compared to the spike,it can be assumed that the stress at ˜(K₂)×FSM_DOL˜(1−3)×FSM_DOL in theparabolic region would not feel the presence of the spike.

This allows the following approximations to be employed:

$\begin{matrix}{{{\sigma_{1}\left( \frac{z}{L} \right)} \cong {{\sigma_{1}^{\prime}\left( \frac{z}{L} \right)} + {\Delta\;{CT}_{spike}}}}{{and},}} & (4) \\{{\sigma_{1}\left( \frac{K_{2} \times {FSM\_ DOL}}{L} \right)} \cong {{\sigma_{1}^{\prime}\left( \frac{K_{2} \times {FSM\_ DOL}}{L} \right)} + {\Delta\;{CT}_{spike}}}} & (5)\end{matrix}$where using the parabolic equation in (1), it is found that:

$\begin{matrix}{{\sigma_{1}\left( \frac{K_{2} \times {FSM\_ DOL}}{L} \right)} = {{{\sigma_{1}(0)}\left\lbrack {{\left( {1 + \frac{1}{n}} \right){\left( {{K_{2} \times \frac{2 \times {FSM\_ DOL}}{L}} - 1} \right)}^{n}} - \frac{1}{n}} \right\rbrack} \cong {{\sigma_{1}^{\prime}\left( \frac{K_{2} \times {FSM\_ DOL}}{L} \right)} + {\Delta\;{CT}_{spike}}}}} & (6) \\{\mspace{79mu}{and}} & \; \\{\mspace{79mu}{{\sigma_{1}(0)} \cong \frac{{\sigma_{1}^{\prime}\left( \frac{K_{2} \times {FSM\_ DOL}}{L} \right)}\mspace{14mu} + {\Delta\;{CT}_{spike}}}{\left\lbrack {{\left( {1 + \frac{1}{n}} \right){\left( {{K_{2} \times \frac{2 \times {FSM\_ DOL}}{L}} - 1} \right)}^{n}} - \frac{1}{n}} \right\rbrack}}} & (7)\end{matrix}$The factor K₂ accounts for nonzero spike stress distribution beyond thedepthDOL_(sp) calculated from the mode spectrum.

It can be demonstrated that if one uses a factor 2 instead of 3 theresults are almost the same, in some cases varying just 1%-3% of σ₁(0).Therefore, if one can find the approximated value of

${\sigma_{1}^{\prime}\left( \frac{3 \times {FSM\_ DOL}}{L} \right)}\mspace{14mu}{or}\mspace{14mu}{\sigma_{1}^{\prime}\left( \frac{1 \times {FSM\_ DOL}}{L} \right)}$in the FSM, formula (6) can be used to compute the original stress ofthe first stress parabola within this range of error.

In practice one can measure approximately

$\sim {{\sigma_{1}^{\prime}\left( \frac{3 \times {FSM\_ DOL}}{L} \right)}\mspace{14mu}{to}}\; \sim {\sigma_{1}^{\prime}\left( \frac{1 \times {FSM\_ DOL}}{L} \right)}$by looking at the stress generated at the transition between guidedmodes and continuum in the spike on Li-glass samples.

This point, where approximately

${\sim {{\sigma_{1}^{\prime}\left( \frac{3 \times {FSM\_ DOL}}{L} \right)}\mspace{14mu}{to}}\mspace{14mu} \sim {\sigma_{1}^{\prime}\left( \frac{1 \times {FSM\_ DOL}}{L} \right)}},$can be used as the point CS_(knee) as shown in FIGS. 3A and 3B as areasonable approximation, that can be measured by computing the distancebetween the mode lines in TM and TE polarization and their refractiveindex. In light of the stress optical coefficient SOC of the materialthe division of the index difference at this point per the SOC wouldlead to the CS_(knee) stress values.

This is in addition to the FSM_DOL and the CS˜σ₂′(0) given by the FSMfor the spike. Therefore CT_(deep)=˜σ₁(0)/n, where for a parabolic deepprofile n=2, and ΔCT_(spike) is given in (3) as (repeated forconvenience)

$\begin{matrix}{{\Delta\;{CT}_{spike}} = \frac{\left( {{\sigma_{2}^{\prime}(0)} - {\sigma_{2}^{\prime}\left( {K_{1} \times {FSM\_ DOL}} \right)}} \right) \times K_{1} \times {FSM\_ DOL}}{T - {K_{1} \times {FSM\_ DOL}}}} & (3)\end{matrix}$

From there one can (repeating the previous equations) then compute thetotal center tension equals the sum of the contributions of the spikeand of the parabolic portion:CT_(tot)=CT_(deep)ΔCT_(spike)  (7)

If desired the depth of compression of the spiked power-law profile canbe calculated/estimated by using the expression:

$\begin{matrix}{{DOL}_{total} = {{DOL}_{deep} - {\frac{{DOL}_{deep}}{n,{CT}}\Delta\;{{CT}_{spike}.}}}} & (8)\end{matrix}$These equations assume that the deep part of the profile is a genericpower profile (parabolic for n=2) in nature and has an added spike nearthe surface. Its validity is better matched when the spike is small instress amplitude and not so deep in comparison to the deeper part of theprofile.

In addition to the generic power ‘n’ profile, the important differencebetween this disclosure and the prior art methods is how the FSM_DOL iscomputed and how the

${CS}_{knee} = {\sigma_{1}^{\prime}\left( \frac{K_{2} \times {FSM\_ DOL}}{L} \right)}$is found using the “last common mode” measured, referring to thehighest-order guided mode that appears both in the TM and the TEspectrum. In an example, if each of the TM has 3 modes and the TEspectrum has 3 modes, then the last common mode is assigned to the thirdmode of each spectrum, when modes are ordered by descending effectiveindex. If the TM spectrum has 3 modes and the TE spectrum has 2 modes,then the last common mode is the second mode in each spectrum when themodes in each spectrum are ordered by descending effective index.

This has direct correspondence to the range of value in which ameasurement is possible with reasonable noise and certainty. This isillustrated in FIG. 7A through 7C, which respectively show example modespectra during the formation of a chemically strengthened glasssubstrate. The mode spectra were taken after diffusion for 1.1 h, (FIG.7A) 2.2 h (FIG. 7B) and 3.8 h (FIG. 7C) in a bath composed of 51 wt %KNO₃ and 49% NaNO₃ at a temperature of 380 C.

The wavelength of the measurement light was 598 nm using a prism couplersystem and camera. It can be observed that, depending on the diffusiontime. a “new mode/fringe” starts to appear at the edge of the screen.This leads to noise in the image and an unstable determination of thetransition between the spike and the long tail of the stress profile.This point is referred as the boundary/continuum or “knee point” due tothe inflection on the stress curve it represents, being illustrated inFIGS. 3A and 3B as the position where the index n^(TE) _(knee) andn_(knee) ^(TM) can be found.

FIG. 7A shows 2 fringes only. FIG. 7B shows a transition region TRbetween the the 2^(nd) and 3^(rd) fringes with a mode appearing in theboundary. FIG. 7C shows 3 fringes only. The concept here is to check thestability of the measurement regardless of accuracy when one uses allmodes plus the boundary/continuum or knee point (known as the “chemicalmode”) in comparison when one measures only using all the known modes inparticular using the ‘last common mode/fringe’ to determine the FSM_DOLand CS_(knee) (the “thermal mode, as described above).

By performing several measurements in a time series of samples describedabove, significant trends can be observed. FIGS. 8A and 8B are plots ofthe depth of layer DOL (μm) versus time (hrs). In the plots, the regionR1 shows the measurement based on two fringes (TM mode) at 589 nm,region R2 shows the transition between using three fringes (TM mode) at589 nm, and region R3 shows the measurement using three fringes (TMmode) at 589 nm. FIGS. 8A and 8B show a time series of measurement ofthe depth of layer DOL of the spike (FSM_DOL) using the aforementioned“chemical mode” or “all mode fringes” plus the separation between lastfringe and knee point are used in computation (FIG. 8A) and the “thermalmode” or “all mode fringes” only are used in computation (FIG. 8B),which is limited by the last fringe.

FIGS. 8A and 8B allow one to clearly observe the regions R1 and R3 thatprovide a relatively stable measurement for FSM_DOL where the measuredFSM_DOL does not oscillate (here there is no mode coming in thecontinuum region). There is however a region R2 where the FSM_DOLoscillates due to the fact that sometimes one computes an extra mode inthe continuum region of the mode spectrum.

For our purposes, regions with 2 or more modes are acceptable but inpractice we are interested in the case for diffusion times of T˜3.5hours as set-point. In this case, one can further see that whenmeasuring using only ‘all the fringes’ and not including the spacingbetween the last known fringe and the continuum (see 54TE, 54TM of FIG.6B), the FSM_DOL measurement has less of a spread (i.e., a smallerstandard deviation). Therefore, it can provide means to control aprocess more robustly and identify the location of these stablemeasurement regions R1 and R4 more efficiently.

FIGS. 9A and 9B are similar to FIGS. 8A and 8B and are plots of thecompressive stress at surface CS (MPa) versus time. The plot of FIG. 9Auses the chemical mode or “all mode fringes” plus the separation betweenlast fringe and the knee point in computation in comparison to FIG. 9B,which uses the thermal mode or “all mode fringes” only, with thecomputation being limited by the last fringe. In this case, largechanges in values are not expected because only the first 2 fringes areused in the computation of the CS, and this is indeed what is observed.

FIGS. 10A and 10B are similar to FIGS. 9A and 9B. FIG. 10A uses thechemical mode or the boundary/continuum or knee point to compute thestress at the knee. FIG. 10B uses the thermal mode or the “last commonmode” and not the knee point to compute the local stress beingmultiplied by a scaling constant to estimate the approximated stress atthe knee.

In this case, it is important to mention that the in the “chemical mode”of the FSM-6000 prism-coupling stress meter, critical angle and itscorresponding effective index are found by the position of theidentified and saved boundary between the TIR region having the discretemodes, and the continuum of radiation modes coupled to the deep region,and the knee stress can be calculated by:

$\begin{matrix}{{\sigma_{knee} = \frac{\left( {n_{crit}^{TE} - n_{crit}^{TM}} \right)}{SOC}},} & (9)\end{matrix}$

The “thermal mode” of the FSM-6000 instrument computes abstract stressvalues corresponding to each mode common to the TM and TE spectrum.These abstract stress values are obtained by dividing the difference ofthe effective indices of the TM and the TE mode in question by thestress-optic coefficient (SOC). The present inventors have determinedthat the abstract stress corresponding to the “last common mode” can beused to compute the stress at the knee, because there is substantialspatial overlap between the spatial distribution of the last mode guidedin the spike, and the region of the knee in the stress profile. In onerelatively crude embodiment, the knee stress can be approximatelyobtained by multiplying the surrogate last-common-mode stress by ascaling factor K₃. This calibration factor is found empirically bycomparing the surrogate stress of the last common mode with the actualknee stress measured by independent means (for example, by therefractive-near-field technique, by polarimetric stress measurements, orby computer simulations of diffusion and the resulting stressdistribution).

The experimental factor K₃ needs to be acquired via measurement at the“knee point” and calculation of the surrogate stress of the last commonmode to generate a scaling that can be used for a particular range ofrecipes.

In the particular case here for diffusion times of about T=3.5 h, thisscaling factor is K₃=0.646. Therefore, using the “last common mode,” onecan compute the stress at the knee and use this information in theprevious formulas as given by:

$\begin{matrix}{\sigma_{knee} = {{K_{3}\frac{\left( {n_{last\_ mode}^{TE} - n_{last\_ mode}^{TM}} \right)}{SOC}} = {\sigma_{1}^{\prime}\left( \frac{K_{2} \times {FSM\_ DOL}}{L} \right)}}} & (10)\end{matrix}$

The last step is to find the K₂ factor. In an example, this is doneexperimentally by measurements of the stress profile by other means(e.g., via destructive measurements) and then comparing to the valuefound using the FSM_DOL. As mentioned before, this value of K₂ isbetween 1 and 3. Therefore K₂ is the scaling of the correct position ofthe knee as a function of the measured FSM_DOL for a certain range ofsamples. As previously mentioned, since the deep part of the profile isslow varying, a certain level of inaccuracy here will not result inlarge errors.

Finally, it is also known that the CS measured by the FSM is anapproximation considering a linear diffusion profile. In some cases, ifa more accurate determination of the CS is needed that can be correctedby another correction factor K₄. This factor is usually quite closeto 1. In practice, it was found that K₄ of about 1.08 leads to moreaccurate representations of the CS in a significant range. Therefore, ifneeded one can also use for more accuracy on CS determinations, therelationship:CS_(corr) =K ₄×CS  (11)

Examples of the use of all the above formulas for the “last know mode”method is set forth in Table 1 in FIG. 11. In Table 1, the formulas wereused to generate all of the critical parameters of the stress profilefor a range of timed glass samples. Constants used were: n=2 (power-lawprofile on parabolic deep part PDP), K₁=1.25, K₂=2, K₃=0.646 and K₄=1(not corrected for CS). The samples were prepared in a bath composed of51 wt % KNO₃ and 49 wt % NaNO₃ at a temperature of 380 C.

In another embodiment of the method, the weight gain of a sample as aresult of ion exchange is used in combination with the prism-couplingmeasurement. The weight gain may be used to verify that enough Na+ ionshave exchanged for Li+ ions such that the use of the parabolic-profilemodel is valid for quality control. For the purpose, a target acceptableweight gain range is prescribed for the ion exchange based on the totalsurface area of the sample and the sample thickness. The weight ofrepresentative samples is measured before and after ion exchange, andthe quality-control prism-coupling measurements are considered valid ifthe measured weight gain per sample falls in the target range.

In another embodiment of the method, advantage is taken of the precisecontrol of the sample shape, and of individual-sample thicknessmeasurements that are common in some production processes. In this caseit is possible to verify that the sample has had adequate weight gain bysimply measuring the sample thickness with high precision (such as +/−1micron), and by measuring the post-ion exchange weight of the sample.From the known shape specification, the measured thickness, and theknown density of the pre-ion-exchanged glass, the weight of thepre-ion-exchanged sample is calculated.

A correction factor may be applied that accounts for a typical volumechange as a result of ion exchange. The weight gain then is estimated bysubtracting from the measured post-ion-exchange weight the estimatedpre-exchanged weight. If the weight gain falls within the target range,the profile is deemed adequately represented by the quality-controlmodel profile, and the prism-coupling QC measurement is consideredvalid.

Another embodiment of the stress-slope method for indirect measurementof CS_(k) offers substantial improvement in the precision of measurementof CS_(k) over the embodiment using the slope of the spike measured fromonly the effective indices of the first two guided modes and the DOL ofthe spike. The original method described above suffered from precisionlimitations associated with normal variability in the detection of thepositions of the fringes in the coupling spectrum corresponding to thesemodes.

The present improved method utilizes three or more modes for at leastone polarization, when available, to calculate the stress slope withsubstantially improved precision, thus allowing much more precisecalculation of CS_(k). The method works well because image-noise-inducederrors in neighboring fringe spacings are anti-correlated, and getsubstantially eliminated when a single linear fit through three or morefringe positions is utilized.

The method substantially improves the precision of the CS_(k)measurement and the CS measurement for a substantially linear spike byusing at least three fringes in at least one of the two polarizations(TM and/or TE) (see FIG. 12 and Tables 2A and 2B, introduced anddiscussed below). Spike shapes that deviate slightly-to-moderately froma substantially linear shape can still benefit an improvement inprecision, although a correction for the shape deviation from linear maybe required in order to obtain the most accurate CS_(k) values. Thistype of correction can be obtained, for example, by a one-timecalibration for each specific spike shape. The calibration may involveidentifying a fraction of the DOL at which CS_(k) is calculated, wherethe fraction can be greater or smaller than 1 (the fraction being equalto 1 for a strictly linear spike).

Method of Calculating Knee Stress

The following describes an example method of calculating the knee stressCS_(k) with reduced susceptibility to the noise of any particular modeby a slope fit method that utilizes several modes at once.

The following equation is used in the method and is for a linear profilethat relates two arbitrary modes m and l confined within the spike,their effective indices being n_(m) and n_(l), and the index slopes_(n):

${\left( {n_{m} + n_{l}} \right)^{\frac{1}{3}}\left( {n_{m} - n_{l}} \right)} = {{s_{n}^{\frac{2}{3}}\left( {\frac{3}{16}\lambda} \right)}^{\frac{2}{3}}\left\lbrack {\left( {{4\; l} + 3} \right)^{\frac{2}{3}} - \left( {{4\; m} + 3} \right)^{\frac{2}{3}}} \right\rbrack}$The above the equation can be used to perform a linear regression, or anevaluation of s_(n) from each pair of modes, and calculate an averagefor s_(n). Mode counting starts from m=0 for the lowest-order mode. Theparameter λ is the optical wavelength used for the measurements.

An example of the method of calculating the knee stress thus includesthe following steps:

-   1) Set a reference index to get all measured modes as actual    effective indices. A good reference index is usually the index    corresponding to the TM critical-angle transition. For Zepler and    FORTE glasses, this index is very close to the original substrate    index, which is usually specified.-   2) Measure all mode effective indices, n_(m), m=0, 1, 2, . . . , for    each polarization, using the angular prism-coupling spectrum of    guided modes.-   3) If desired, assume that n_(m)+n_(l) hardly changes, and assign it    as a constant equal to 2n.-   4) For each pair of integers m, l≥0, calculate

$B_{ml} = {\frac{\left( {\frac{3}{16}\lambda} \right)^{\frac{2}{3}}\left\lbrack {\left( {{4\; l} + 3} \right)^{\frac{2}{3}} - \left( {{4\; m} + 3} \right)^{\frac{2}{3}}} \right\rbrack}{\left( {n_{m} + n_{l}} \right)^{\frac{1}{3}}} \approx \frac{\left( {\frac{3}{16}\lambda} \right)^{\frac{2}{3}}\left\lbrack {\left( {{4\; l} + 3} \right)^{\frac{2}{3}} - \left( {{4\; m} + 3} \right)^{\frac{2}{3}}} \right\rbrack}{\left( {2\;\overset{\_}{n}} \right)^{\frac{1}{3}}}}$

-   5) Perform a linear regression of the equation    y_(ml)≡n_(m)−n_(l)=SB_(ml), to find a capital slope S.-   6) If desired, check if the quality of the linear regression is    adequate (e.g., R² is higher than a minimum requirement).-   7) Find the index slope using s_(n)=S^(3/2)-   8) Find the surface index for the purposes of the subsequent    knee-stress calculation using:

$s_{n} = {\frac{16}{3\left( {{4\; m} + 3} \right)\lambda}\left( {n_{surf} + n_{m}} \right)^{\frac{1}{2}}\left( {n_{surf} - n_{m}} \right)^{\frac{3}{2}}}$$n_{surf} = {n_{m} + {\frac{s_{n}^{\frac{2}{3}}}{\left( {2\;\overset{\_}{n}} \right)^{\frac{1}{3}}}\left\lbrack \frac{3\left( {{4\; m} + 3} \right)\lambda}{16} \right\rbrack}^{\frac{2}{3}}}$

-   9) If higher accuracy is desired, replace 2n, for each pair of modes    with the actual sum of the measured values, n_(m)+n_(l), as    mentioned earlier-   10) For the calculation of 2n, used in the calculation of n_(surf),    optionally use an iterative procedure, where in the first step we    use 2n₀, and on the second iteration, use n₀+n_(surf) ⁽⁰⁾, e.g., use    the estimated surface index form the first iteration to calculate    the average of the surface and the first mode. For faster    calculation, use:    2 n≈n ₀ +n ₀+1.317(n ₀ −n ₁)≡2n ₀+1.317(n ₀ +n ₁)-   11) Calculate the surface CS for the purposes of finding the knee    stress: CS=(n_(surf) ^(TE)−n_(surf) ^(TM)/SOC

$s_{\sigma} = \frac{s_{n}^{TE} - n_{n}^{TM}}{SOC}$

-   12) Calculate the stress slope for the purposes of finding the knee    stress:-   13) Find the DOL from the upper TM spectrum for higher precision-   14) Calculate the knee stress: σ_(knee)=CS+s_(σ)×DOL-   15) If the deeper end of spike differs somewhat from linear as    truncated on the deep side, then apply a correction factor:    σ_(knee)=CS+F×s_(σ)×DOL, where F is the correction factor, usually    between about 0.4 and 1, but for spikes having regions of negative    curvature it could exceed 1. The correction factor can be calculated    by accounting for the actual concentration profile of potassium (K)    measured by secondary-ion mass spectroscopy (SIMS), glow-discrharge    emission spectroscopy (GDOES), or electron microprobe, or it can be    found empirically by comparing measured knee stresses to the    equation above and fitting the value of F that makes them agree.

Clearly the above method can be applied to either or both of the TM andTE index profiles of the potassium-enriched spike, to improve theprecision of CS and CS_(k). The improvement is most significant when itis applied to both the TM and the TE spectra, but it could be used incases where one of the spectra only has 2 guided modes (for example theTE spectrum), in which case the linear regression is applied only to thespectrum having at least 3 guided modes. Furthermore, it can clearly beapplied using in general a different number of TM and TE modes, althoughthe accuracy might be highest when the same number of TM and TE modesare used.

The data from application of the two major embodiments of the slopemethod for indirect CS_(k) calculation to actual prism-couplingmeasurements of several samples covering a range of different DOL areshown in Tables 2A and 2B, below. Table 2A shows the results of theprior-art method of calculation employing two modes while Table 2B showsthe results of the improved method of calculation as disclosed hereinthat uses additional modes.

TABLE 2A Single mode pair stress slope method CS CS_(K) 528.2 143.9519.1 136.2 520.4 130.3 515.7 126.7 512.9 112.9 519.3 122.3 509.5 121.2513.1 117.9 517.7 119.7 514.2 120.8 515.3 123.1 515.6 127.4 515.9 118.5517.8 128.7 515.5 125.0 Standard Deviation  4.3  7.8

TABLE 2B Single mode pair stress slope method CS CS_(K) 525.4 144.6516.4 142.5 517.5 146.0 513.2 143.8 509.8 149.1 516.4 148.3 506.9 142.6511.3 144.6 514.8 148.7 512.1 144.9 512.9 144.8 513.1 143.3 513.8 146.6515.4 144.1 513.3 143.9 Standard Deviation  4.1  2.1

From the data of Tables 2A and 2B, plots of CS vs extracted CS_(k) usingthe two methods from first two modes only (fitted curve A), and fromusing all available modes for slope calculation (fitted curve B) areshown in FIG. 12. The last row in Table 2B shows a substantially reducedstandard deviation in the indirect measurement of CS_(k) when the slopeis extracted from using all the modes (three or more), rather than onlythe two lowest-order modes as used in the method of Table 2A (fittedcurve A).

The data in FIG. 12 for the same improved method has a substantiallysmaller spread for CS_(k), indicative of the reduced standard deviation.The data also shows a much smaller dependence on CS, suggesting that theextraction of CS_(k) from only the first two fringes is subject tomeasurement errors that tend correlate CS_(k) with CS.

Two other embodiments of the method offer a substantial improvement inthe accuracy of measurement of CS_(k) based on the other indirect methoddisclosed earlier, i.e., the method that uses the birefringence of thehighest-order guided mode of the spike to estimate CS_(k). Thehighest-order guided mode has effective index only slightly higher thanthe effective index corresponding to the depth at which the knee of thestress profile occurs. Thus, the birefringence of that mode issignificantly affected by the knee stress. If the spike CS and DOL arekept constant, then the knee stress CS_(k) would be essentially the soledriver of changes in the birefringence of the highest-order spike mode.

The method described above calculates the knee stress CS_(k) as afraction of the birefringence of the highest-order spike mode. A problemwith this method can occur when the CS and DOL of the spike are allowedto vary moderately or significantly by a relatively broad productspecification, as typical for chemically strengthened cover glasses.

The two improved embodiments of the method for calculating the kneestress CS_(k) disclosed below correct for the effects of varying CS andDOL of the spike on the birefringence of the surrogate guided mode sothat the indirectly recovered value of CS_(k) is more accurate.Improvement of the accuracy of CS_(k) measurements is sought bycorrecting for significant distortions of indirectly-extracted CS_(k)values by the last-fringe method (birefringence of the highest-orderguided mode acting as a surrogate for the knee-stress-inducedbirefringence).

In one aspect of the method, a derivative of the birefringence of thechosen surrogate guided mode is calculated with respect to deviations ofthe CS, DOL, and CS_(k) from their nominal values for the targetproduct. Then CS_(k) is calculated from the measured surrogate-modebirefringence, after applying corrections associated with the product ofthese calculated or empirically extracted derivatives, and thecorresponding measured deviations of CS and DOL from the target values.

In an example, the spike shape may be assumed to have a lineardistribution from the surface to the depth of the knee. This is a goodapproximation for a single-step process. An erfc-shaped spike can beconsidered a good approximation for a two-step process, where thefirst-step uses a lower substantially nonzero potassium concentration inthe bath, and forms a substantially lower CS than the second step, andwhere the second step has a substantially shorter ion exchange time atapproximately the same or lower temperature than the first step. Thespecific shape of the profile does not affect the method of correction,only the absolute values of the correction factors.

In the present example, the last-fringe birefringence was calculated byusing the linear-spike approximation. The fabrication process involves asample of 0.5 mm thick Corning 2321 glass subjected to ion exchange at380 C for approximately 1.6 hours in a mixture having approximately 20%NaNO₃ and 80% KNO₃ by weight. The nominal CS for the target is 675 MPaand the nominal DOL is 9 microns.

Table 3 is presented in FIG. 13 and shows the calculated effectiveindices of the three guided modes for both the TM and TE polarizationsfor several different assumed values of CS_(k), CS, and DOL. Effectiveindices can be calculated numerically, e.g., by a mode solver thatnumerically solves the wave equation, or by a transfer-matrix approach,for example. Such methods are well known in the art.

The eighth column shows the birefringence of the third guided mode (modeindexing counts from 0, so the third guided mode is TM2/TE2). The ninthcolumn shows the abstract compressive stress CSn2 corresponding to thebirefringence of the highest-order guided mode (in this case, thethird). This abstract compressive stress is obtained by dividing themode birefringence by the stress-optic coefficient SOC.

The rightmost column shows the calculated change in the calculatedabstract compressive stress by a unit change in the correspondingparameter (i.e., a 1 MPa change in CS_(k), a 1 MPa change in surface CS,or a 1 micron change in DOL). These can be used approximately as thederivatives of the abstract compressive stress with respect to changesof the driving parameters. It can be seen from Table 3 that theso-calculated derivatives may be slightly different on the side ofincreasing a parameter than on the side of decreasing of the sameparameter. This is due to using a finite interval for calculating thederivatives. The difference can be decreased if a smaller interval isused for the estimates. In practice, the average derivative from thepositive and negative side of the parameter change may be used over theentire interval to provide a fairly good correction.

If the surrogate abstract mode compressive stress calculated from thebirefringence of the highest-order guided common mode is labeledCS_(sur), then the corrected value of knee stress can be calculatedusing the measured values of CS, DOL, and CS_(sur), and using thenominal values for CS, DOL, CS_(k) and CS_(sur). Generally, thecalculation can use the form

${CS}_{k} = {{CS}_{k}^{nom} + \frac{\left( {{CS}_{sur} - {CS}_{sur}^{nom}} \right) - {CorrCS} - {CorrDOL}}{\frac{{dCS}_{sur}}{{dCS}_{k}}}}$where the corrections CorrCS and CorrDOL are calculated from the productof deviations of CS_(sp) and DOL_(sp) from their nominal values, and thecorresponding sensitivities of the surrogate stress CS_(sur) to changesin CS_(sp) and DOL_(sp). Note that in the present disclosure, when CS isused without any subscript, it means the surface compressive stress ofthe spike CS_(sp).A simple embodiment of the above method is using the equation:

${CS}_{k} = {{CS}_{k}^{nom} + \frac{\begin{matrix}{\left( {{CS}_{sur} - {CS}_{sur}^{nom}} \right) - {\left( {{CS} - {CS}^{nom}} \right)\frac{{dCS}_{sur}}{dCS}} -} \\{\left( {{DOL} - {DOL}^{nom}} \right)\frac{{dCS}_{sur}}{dDOL}}\end{matrix}}{\frac{{dCS}_{sur}}{{dCS}_{k}}}}$

In the above example, the equation reduces to:

${CS}_{k} = {95 + \frac{\left( {{CS}_{sur} - 348.2} \right) - {\left( {{CS} - 650} \right)0.76} - {\left( {{DOL} - 9.0} \right)47.7}}{0.525}}$The above use of linear relationship between the deviations of CS_(sp)and DOL_(sp) from their nominal values, and the correspondingcorrections CorrCS and CorrDOL makes CS_(k) susceptible to increasedstandard deviation when the measurements of CS_(sp) and/or DOL_(sp) aresubject to substantial random error (noise). In some cases thisincreased standard deviation can be problematic. Limiting the amount ofcorrection by using a nonlinear relationship between each correction andthe corresponding deviation in CS_(sp) or DOL_(sp) from its nominalvalue can help stabilize the calculated CS_(k). In an example, thecorrections can be calculated by the following:

${CorrCS} = {\Delta_{1} \times {\tanh\left( \frac{\frac{{dCS}_{sur}}{{dCS}_{sp}}\left( {{CS}_{sp} - {CS}_{sp}^{nom}} \right)}{\Delta_{1}} \right)}}$And${CorrDOL} = {\Delta_{2} \times {\tanh\left( \frac{\frac{{dCS}_{sur}}{{dDOL}_{sp}}\left( {{DOL}_{sp} - {DOL}_{sp}^{nom}} \right)}{\Delta_{2}} \right)}}$Where Δ₁ and Δ₂ are limiting values of the corrections, preventingover-compensation due to noise in the CS_(sp) and DOL_(sp) values.

In another embodiment of the method, the factor K₃ used to relate thesought knee stress CS_(k) and surrogate stress (calculated from thebirefringence of the last guided mode), is allowed to vary with thesurface CS and the spike DOL, so that the extracted value of CS_(k) frommeasurements of the surrogate stress can better match the actual kneestress over a variety of CS and DOL combinations.

In an example, the CS and DOL were varied slightly in simulations of theoptical modes of a chemically strengthened sample with the knee stressin the vicinity of 150 MPa, CS in the vicinity of 500 MPa, and DOL inthe vicinity of 10 microns. The knee stress, which was input in thesimulations, was then divided by the surrogate abstract mode stress thatwas calculated by the simulation, to find how the factor K₃ varied withCS and DOL.

FIG. 14A shows a calculated dependence of K₃ on CS (i.e., K₃ vs. CS(MPa)). The vertical lines show a range of CS over which K₃ can beapproximated as constant for the purposes of CS_(k) calculations in thisexample. In other cases with substantially steeper spikes (e.g., slopein the vicinity of 100 MPa/micron), the range of CS over which K3 can beapproximated as constant would be narrower.

FIG. 14B shows the derivative of K₃ with respect to CS (dK₃/dCS)calculated from the same data. It can be used to calculate a K₃ valuefrom a nominal K₃ value obtained during a calibration measurement of astress profile (e.g., by polarimetry or refractive-near-fieldmeasurements) by applying a correction.

FIG. 15 shows the dependence of K₃ on the DOL. At smaller DOL valuesthan about 9.5 microns the dependence becomes steeper, presumably due tothe third TE mode transitioning from a guided mode to a leaky mode.Usually the measurement is not in the sweet spot in that case, and sucha region would be avoided.

FIG. 16 shows a derivative of the scaling factor K₃ with respect to DOL(dK₃/dDOL) having a region of relatively small and little changingderivative, and a region of fast-changing derivative, growingsubstantially in absolute value. Operating in the region where thederivative of K₃ with respect to DOL is small by absolute value andchanging little is preferred for the embodiments involving indirectmeasurements of CS_(k) based on the birefringence of the last guidedmode.

In an example, the corrected value of K3 can be calculated as follows:

${K_{3}\left( {{CS},{DOL}} \right)} = {{K_{3}^{nom}\left( {{{CS} = {CS}^{nom}},{{DOL} = {DOL}^{nom}}} \right)} + {\left( {{CS} - {CS}^{nom}} \right)\frac{{dK}_{3}}{dCS}} + {\left( {{DOL} - {DOL}^{nom}} \right)\frac{{dK}_{3}}{dDOL}}}$

In another example, the value of K₃ can be tabulated for a matrix of CSand DOL combinations, and read out during measurements by an algorithmselecting the closest CS/DOL combination to the measured values of CSand DOL.

In another embodiment of the method, the value of K₃ need not becorrected. Instead, the range of combinations of CS, DOL, anduncorrected CS_(k) can be separated in several regions, such thatcombinations having high CS and DOL, and low CS_(K) can be rejectedduring quality-control measurements. This account for the observationthat high CS and DOL both tend to raise the indirectly-measured CS_(K)by the highest-guided-mode surrogate method.

In one example, a process space (process window) is defined by theproduct of the CS and DOL specifications. This process space is thensplit into two or more regions, preferably in parallel to the diagonalrelating the point (CSmax, DOLmin) with the point (CSmin, DOLmax). Thenfor each region, a different lower limit of CS_(K) is used as a reasonto reject a part, with the so required CS_(K) lower limit generallyincreasing with increasing CS and increasing DOL. In another example,the CS/DOL process space can be split into two or more sub-regions bycurves corresponding to the condition CS*DOL=const, or (CS−CS_(K)^(nom))*DOL=const.

It will be apparent to those skilled in the art that variousmodifications to the preferred embodiments of the disclosure asdescribed herein can be made without departing from the spirit or scopeof the disclosure as defined in the appended claims. Thus, thedisclosure covers the modifications and variations provided they comewithin the scope of the appended claims and the equivalents thereto.

What is claimed is:
 1. A method of characterizing a stress profile of achemically strengthened glass substrate formed by the in-diffusion ofalkali ions and having an upper surface and a body, a shallow spikeregion of stress immediately adjacent the upper surface and a deepregion of slowly varying stress within the body and that intersects thespike region at a knee, wherein the method comprises: measuring a TMmode spectrum and a TE mode spectrum of the glass substrate, wherein theTM mode spectrum and the TE mode spectrum each include mode lines and atransition associated with a critical angle; determining a surfacecompressive stress CS_(sp) of the spike using the TM and TE modespectra, or a measurement of a surface concentration of at least onetype of the alkali ions; measuring a difference between the TE and TMtransition locations to determine an amount of birefringence BR; andcalculating the knee stress as CS_(knee)=(CFD)(BR)/SOC, where SOC is thestress-optic coefficient and where CFD is a calibration factor.
 2. Themethod according to claim 1, further comprising determining that a depthof layer DOL_(sp) of the spike is within a desired range by confirmingthat at least one of the TM and TE spectra includes at least apredetermined number of fringes.
 3. The method according to claim 2,wherein the predetermined number of fringes is
 2. 4. The methodaccording to claim 2, wherein the determining that the depth of layerDOL_(sp) of the spike is within a desired range comprises confirmingthat the TM spectrum includes at least a predetermined number offringes.
 5. The method according to claim 1, wherein the CFD is between0.5 and 1.5.
 6. The method according to claim 1, wherein the alkali ionsare Na and K, wherein the glass substrate contains Li, and wherein thedeep region of the profile is enriched with Na and the shallow region isenriched in K.
 7. The method according to claim 1, further comprisingapproximating a stress profile of the deep region by a power law havinga power coefficient between 1.3 and
 4. 8. The method according to claim1, wherein the TM mode spectrum and the TE mode spectrum each have afractional part of a mode number between 0.2 and 0.6.
 9. A method ofcharacterizing a stress profile of a chemically strengthened glasssubstrate formed by diffusion of alkali ions and having an upper surfaceand a body, a shallow spike region of stress immediately adjacent theupper surface, and a deep region of slowly varying stress within thebody and that intersects the spike region at a knee, wherein the methodcomprises: measuring a TM mode spectrum and a TE mode spectrum of theglass substrate, wherein the TM mode spectrum and the TE mode spectrumeach include mode lines associated with the spike region; a portion oftotal-internal reflection (TIR), a portion of partial reflection wherethere is optical coupling into the deep region of the substrate, and thetransition between the two portions that corresponds to a criticalangle; determining a surface compressive stress CS_(sp) of the spikeusing either at least one of the TM and TE mode spectra or a measurementof a surface concentration of at least one type of alkali ion thatresides adjacent the surface; estimating a stress-induced birefringenceBR given by:BR=n _(LM) ^(TE) −n _(LM) ^(TM) where n_(LM) ^(TE) is the effectiveindex of a TE spike mode of the highest-common-order, n_(LM) ^(TM) isthe effective index of TM spike mode of the highest-common-order; andcalculating the knee stress as CS_(knee)=(CFD)(BR)/SOC, where SOC is thestress-optic coefficient and where CFD is a calibration factor.
 10. Themethod according to claim 9, further comprising determining that a depthof layer DOL_(sp) of the spike is within a desired range by confirmingthat at least one of the TM and TE spectra includes at least apredetermined number of fringes.
 11. The method according to claim 10,wherein the predetermined number of fringes is
 2. 12. The methodaccording to claim 10, wherein the determining that the depth of layerDOL_(sp) of the spike is within a desired range comprises confirmingthat the TM spectrum includes at least a predetermined number offringes.
 13. The method according to claim 9, wherein the CFD is between0.5 and 1.5.
 14. The method according to claim 9, where the alkali ionsare Na and K, wherein glass substrate contains Li, and wherein the deepregion of the profile is enriched with Na and the shallow region isenriched in K.
 15. The method according to claim 9, wherein the TM modespectrum and the TE mode spectrum each have a fractional part of a modenumber between 0.2 and 0.6.
 16. A method of characterizing a stressprofile of a chemically strengthened glass substrate formed by diffusionof alkali ions and having an upper surface and a body, a shallow spikeregion of stress immediately adjacent the upper surface and a deepregion of slowly varying stress within the body and that intersects thespike region at a knee, wherein the method comprises: measuring a TMmode spectrum and a TE mode spectrum of the glass substrate, wherein theTM mode spectrum and the TE mode spectrum each include mode lines and atransition that corresponds to a critical angle; determining a surfacecompressive stress CS_(sp) of the spike using either at least one of theTM and TE mode spectra or a measurement of a surface concentration of atleast one type of alkali ion that resides adjacent the surface;measuring a difference between the TE and TM transition locations todetermine an amount of birefringence BR, wherein BR is given by:BR=n _(LM) ^(TE) −n _(LM) ^(TM) where n_(LM) ^(TE) is the effectiveindex of a highest-common-order TE spike mode, n_(LM) ^(TM) is theeffective index of a highest-common-order TM spike mode; and calculatinga knee stress as CS_(k) using the equation:CS_(k) =K ₃ ×BR/SOC where SOC is the stress-optic coefficient, and K₃ isa calibration factor that is in the range between 0.2 and
 2. 17. Themethod according to claim 16, further comprising determining that adepth of layer DOL_(sp) of the spike is within a desired range byconfirming that at least one of the TM and TE spectra includes at leasta predetermined number of fringes.
 18. The method according to claim 17,wherein the predetermined number of fringes is
 2. 19. The methodaccording to claim 17, wherein the determining that the depth of layerDOL_(sp) of the spike is within a desired range comprises confirmingthat the TM spectrum includes at least a predetermined number offringes.
 20. The method according to claim 16, where the alkali ions areNa and K, wherein glass substrate contains Li, and wherein the deepregion of the profile is enriched with Na and the shallow region isenriched in K.
 21. The method according to claim 16, wherein the TM modespectrum and the TE mode spectrum each have a fractional part of a modenumber between 0.2 and 0.6.